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The maximal cardinality of a code W on the unit sphere in n dimensions
with (x, y) ≤ s whenever x, y ∈ W, x 6= y, is denoted by A(n, s). We use two
methods for obtaining new upper bounds on A(n, s) for some values of n and s.
We find new linear programming bounds by suitable polynomials of degrees which
are higher than the degrees of the previously known good polynomials due to
Levenshtein [11, 12]. Also we investigate the possibilities for attaining the Levenshtein
bounds [11, 12]. In such cases...
We obtain new combinatorial upper and lower bounds for the
potential energy of designs in q-ary Hamming space. Combined with results
on reducing the number of all feasible distance distributions of such designs
this gives reasonable good bounds. We compute and compare our lower
bounds to recently obtained universal lower bounds. Some examples in the
binary case are considered.
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